Problem: Factor the following expression: $7$ $x^2+$ $18$ $x+$ $8$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(7)}{(8)} &=& 56 \\ {a} + {b} &=& & & {18} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $56$ and add them together. The factors that add up to ${18}$ will be your ${a}$ and ${b}$ When ${a}$ is ${4}$ and ${b}$ is ${14}$ $ \begin{eqnarray} {ab} &=& ({4})({14}) &=& 56 \\ {a} + {b} &=& {4} + {14} &=& 18 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {7}x^2 +{4}x +{14}x +{8} $ Group the terms so that there is a common factor in each group: $ ({7}x^2 +{4}x) + ({14}x +{8}) $ Factor out the common factors: $ x(7x + 4) + 2(7x + 4) $ Notice how $(7x + 4)$ has become a common factor. Factor this out to find the answer. $(7x + 4)(x + 2)$